What pushed the United States to enter World War I, and how did its entry affect the outcome of the war?

If you are asking what the final straw was that caused the U.S. to declare war on Germany and enter WWI on the Allied side, the answer is threefold: 1) the German sinking of the Lusitania, the Arabic, and the Sussex; 2) the Zimmerman Note; and 3) the resumption of unrestricted submarine warfare by the Germans.

In 1915, the Germans were using unrestricted submarine warfare to torpedo and sink passenger liners, killing civilians. Both the Lusitania, with 128 Americans on board, and the Arabic were attacked without warning. In April of 1916 Germany attacked the Sussex, killing several more Americans. At this point, Woodrow Wilson gave the Germans an ultimatum that condemned their use of submarines to attack merchant vessels trading with the Allies. Wilson warned the German government that the United States would act against Germany if they did not revoke the policy of unrestricted submarine warfare. Germany’s Foreign Minister, Gottlieb von Jagow, revoked the policy and promised that future sinking would ensue only after clear warning.

In January, 1917, the German Foreign Minister, Authur Zimmerman, sent a telegram (the Zimmerman Note) to the German Minister in Mexico encouraging an alliance with the Central Powers if Germany entered into war with the United States. Sweetening the pot for Mexico, Zimmerman offered US territory as spoils of the presumed victory. The note, however, was intercepted by the British who forwarded the message to President Woodrow Wilson. After Mexican President Venustiano Carranza determined it improbable that Mexico would benefit from entering a war against the U.S., Wilson recognized the Carranza government as a political gesture.

After removal of the threat of Mexica to the U.S., Germany chose to renew its unrestricted submarine warfare on February 1, 1917.

Germany's rejection of the ultimatum gave Wilson no choice but to declare war on Germany and the Central Powers. The Zimmerman note was icing on the cake.

If the United States had not entered the war, even as late as it did, then Germany and the Central Powers MAY have been able to win the war. Historians disagree on the real effect. However, the US entry into the war, through a large contingent of fresh troops, helped the Allies in their quest to overcome Kaiser Wilhelm. The Brits and the Germans were both worn out and morale was sagging. Fresh troops tipped the advantage in British favor, helping the Allies to win the war.